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RoseWind [281]
3 years ago
11

Evaluate the surface integral S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of

F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = xy i + yz j + zx k S is the part of the paraboloid z = 8 − x2 − y2 that lies above the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and has upward orientation
Mathematics
1 answer:
Maru [420]3 years ago
6 0

Parameterize S{/tex] by[tex]\vec s(u,v)=u\,\vec\imath+v\,\vec\jmath+(8-u^2-v^2)\,\vec k

with 0\le u\le1 and 0\le v\le1.

Take the normal vector to S to be

\vec s_u\times\vec s_v=2u\,\vec\imath+2v\,\vec\jmath+\vec k

Then the flux of \vec F across S is

\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\int_0^1\int_0^1\vec F(x(u,v),y(u,v),z(u,v))\cdot(\vec s_u\times\vec s_v)\,\mathrm du\,\mathrm dv

=\displaystyle\int_0^1\int_0^1(uv\,\vec\imath+v(8-u^2-v^2)\,\vec\jmath+u(8-u^2-v^2)\,\vec k)\cdot(2u\,\vec\imath+2v\,\vec\jmath+\vec k)\,\mathrm du\,\mathrm dv

=\displaystyle\int_0^1\int_0^1\bigg(2u^2v+(u+2v^2)(8-u^2-v^2)\bigg)\,\mathrm du\,\mathrm dv=\boxed{\frac{1553}{180}}

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\boxed {-\frac{12}{5}}

Step-by-step explanation:

Solve the following expression:

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-\frac{128}{2187} \times (\frac{3}{5})^{6} \times (\frac{5}{6})^{5} \times (\frac{1}{3})^{-7}

-Calculate \frac{3}{5} to the power of 6:

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-\frac{128}{2187} \times (\frac{729}{15625}) \times (\frac{5}{6})^{5} \times (\frac{1}{3})^{-7}

-Multiply both -\frac{128}{2187} and \frac{729}{15625}:

-\frac{128}{2187} \times (\frac{729}{15625}) \times (\frac{5}{6})^{5} \times (\frac{1}{3})^{-7}

-\frac{128}{46875} \times (\frac{5}{6})^{5} \times (\frac{1}{3})^{-7}

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-\frac{128}{46875} \times (\frac{3125}{7776}) \times (\frac{1}{3})^{-7}

-Multiply both -\frac{128}{46875} and \frac{3125}{7776}:

-\frac{128}{46875} \times (\frac{3125}{7776}) \times (\frac{1}{3})^{-7}

-\frac{4}{3645} \times (\frac{1}{3})^{-7}

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-\frac{4}{3645} \times 2181

-Multiply both the -\frac{4}{3645} and 2187:

-\frac{4}{3645} \times 2181

\boxed {-\frac{12}{5}}

Therefore, the final answer is -\frac{12}{5}.

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